[Math] describe the conjugacy classes of an abelian group

Question

describe the conjugacy classes of an abelian group

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each class is singleton set as for abelian group $ax=xa$ $\forall x \in G$ which gives $xax^{-1}=a$

Am I right?

Answer 1

Noting to say more than @Tobias's comment for your problem. Just noting that if we set $$\Delta(x)=\{x^g\mid g\in G\}=\{g^{-1}xg\mid g\in G\}$$ then we can see that $|\Delta(x)|=|G:C_G(x)|$ and since $G$ is abelian so every conjugacy classes is singleton .